Topological optimality condition for the identification of the center of an inhomogeneity

The inverse scattering problem for inhomogeneous media is considered within the topology optimization framework. Varying the complex-valued refractive index we derive a zero-order necessary optimality condition in minimizing the L2 misfit cost functional of the far-field measurement. The topology asymptotic expansion of the optimality condition leads to an imaging operator, which is used to identify the center of the unknown inhomogeneity using few farfield measurements. Numerical tests show high precision and stability in the reconstruction using our optimality condition based imaging both in two and three dimensions.